Boundary Stabilization of the Navier--Stokes Equations in the Case of Mixed Boundary Conditions

Abstract

We study the boundary feedback stabilization, around an unstable stationary solution, of a two dimensional fluid flow described by the Navier--Stokes equations with mixed boundary conditions. The control is a localized Dirichlet boundary control. A feedback control law is determined by stabilizing the linearized Navier--Stokes equations around the unstable stationary solution. We prove that the linear feedback law locally stabilizes the Navier--Stokes system. Thus, we extend results previously known in the case when the boundary of the geometrical domain is regular and when only Dirichlet boundary conditions are present to the case with mixed boundary conditions.